Topology and its Applications 11 (1980) 37-46 @ North-Holland Publishing Company

AN INJECTIVE CHARACTERIZATION OF’ PEANO SPACES

Burkhard HOFFMANN Mathematical Institute, Oxford, England

Received 20 November 1978 Revised 26 April 1979

It is shown that a continuous map defined on a closed zero-dimensional sabspace S of a 7’ into a Peano space X can be continuously extended over T or, equivalently, X is an AE(0, OO),and this property precisely characterizesPeano spaces within the class of compact metric spaces. Surjectively, a compact AE(0, ~a)of arbitraryweight is proved to be the continuous image of a Tychonoff cube by a map satisfying the zero-dimensional lifting property.

AMS (MOS) Subj. Class. (1970): Primary 54F25,54ClO

Peano space zero-dimensional lifting propel :y AE(m, n)

0. Introduction

Locally connected, connected metric spaces or Peano spaces were characterized by Hahn and Mazurkiewicz in their celebrated theorem as the continuous images of the unit interval. Although considerable effort has been made in various directions to generalize their theorem to non-met&able spaces (see e.g. [ 111, [ 121 and their references), no satisfactnry solution has yet been fcund. Combining the standard technique of the proof of the Hahn-Mazurkiewicz theorem with the concept of zero-dimensional lifting property or z.d.1.p. we prove that, in the category of compact spaces and continuous maps, Peano spaces are precisely the metrizable absolute extensors for zero- to infinite-dimensional spaces (Theorem 2.2). This result furnishes us with an injective characterization of Peano spaces which seems to be suitable for a generalization to compact spaces of uncountable weight. Theorem 2.1 provides a surjective characterization of this generalized class. For pairs (m, n) of extended integers,, 0 G m G n s 00, a compact space X is defined to be an ahlute extensor for m- to n-dimensional spaces or AE(m, n), if for any embedding C:S + T of compact spaces S, T with dim S G m, dim T 6 n and any continuous map t# : S +X there exists a continuous map $: T -) X such that $0 c = 4. In [4], R. Haydon proved the equivalence of the notions of Dugundji space and AE(0,0). Obviously, a compact space is an absolute retract if and only if it is an AE@, 00). As a hybrid of these two extreme cases, we characterize the class of Peano

37 38 B. Hoffmann / An injective characterization of Peano spaces spaces as the class of metrizable AE(0, a). Furthermore, without restriction to metrizabillity an AE(0,00) is surjectively classified as a continuous image of a Tychonoff cube by a map satisfying the z.d.1.p. (Theorem 2.1). This surjective characterization of AE(0,00) should be contrasted with the characterization given in [S] for AE(0,0) as the continuous image of a Cantor cube be a map satisfying the z.d.1.p. Recall that, as a consequence of Milutin’s lemma, a compact .metrizable space is the continuous image of the by a map allowing a regular averaging operator or r.a.o. (see [8], Theorem 5.6). Applying results of a previous paper [S], we deduce here that a Peano space is a continuous image of the unit interval by a map allowing an r.a.o. (Theorem 2.2). This article is devided into two sections. In Section 1 we build up a machinery concerning the concept of z.d.1.p. and culminating in Proposition 1.6 which, in turn, is used as our main tool for the proofs in Section 2. We like to point out that the results and proofs contained in this paper are of a purely topological nature. The reader who is not familiar with the algebraic notion of r.a.o. may choose to substitute it by the topological notion of z.d.1.p. whenever the compact spaces involved are metrizable. For the equivalent of the notions of r.a.o. and z.d.1.p. within the realm of compact metrizable spaces was obtained in a previous article [S].

Notations The closed unit interval of the reals is denoted by I and D is the discrete two-point space (0, 1). We follow the convention of [lo] in identifying a cardinal number with the corresponding initial ordinal. Hence the first two infinite cardinals are indicated by the symbols or)and ml, A countable product of copies of D is homeomorphic to Do. The space R”‘ is frequently identified with the Cantor set, consisting of those elements of I that have a triadic expansion in which the digit 1 does not occur. The dimension function dim indicates, as usual, the covering dimension. Note that dim X .= -1 means X = 0. The weight of a compact space X is the smallest cardinal r such that t&ere exists a base for the topology of X of cardinality T. We can embed X in the Tychonoff cube I’, For a compact space Z denote by C(Z) the Banach space of continuous real- valued functions of 2, equipped Gth the supremum norm. Suppose we are given a continuous surjection 8: X + Y ck compact spaces X, Y. A linear map u: C(X) * C(Y) is said to be a regular aver aging operator or r.a.0, for 8, if u is positive and u( f 00) = f holds for all elements 1; of C( Y). Necessarily, an r.a.0. is of norm 1.

1. The concept of zero-dimensional Ming property

The fundamental tool for this piece of research is provided by the notion of zero-dimensional lifting property which we introduced in [S]. For the convenience of the reader we record it here again. B. Hoffmann / An injective characterization of Peano spaces 39

Definition 1.1. Let X, Y be compact spaces and 8: X + Y a continuous surjection. Then 0 satisfies the zero-dimensional lifting property or z.dJ.p., if for any zero- dimensional compact space T and for any continuous map 4: T + Y there exists a continuous lifting a: T + X such that the diagram

commutes, i.e. 80 c = 4.

Let us start the investigation of the notion of z.d.1.p. by compiling some elementary observations. Trivially, any retraction of compact spaces satisfies the z.d.1.p. and if the image space of a continuous surjection is zero-dimensional the converse holds as well. Any projection of a product of compact spaces onto one of its factors satisfies the z.d.1.p. The notion of z.d.1.p. is stable under finite compositions. Lifting maps coordinatewise immediately yields that the product map

(e,)a,A: n xa+ n yk (Xa)acAH(oa(xa))aEA9 aeA aEA

satisfies the z.d.l.p., if each coordinate map 6a : Xa + Ya has this property. In [SJ, we proved that for a continuous surjection of compact spaces satisfying the z.d.1.p. entails the existence of an r.a.o. Moreover, it was shown that the converse holds, if the compact spaces involved are metrizable. Hence, in particular, an open continuous surjection of compact metrizable spaces satisfies the z.d.l.p., since it allows an r.a.o. by a result of Pelczyriski [8, p. 651. In the sequel, we shall utilize a lemma which expresses the stability of the notion of z.d.1.p. under the operations of restriction and extension. Its straightforward verification is left to the reader.

Lemma 1.2. Let 8: X + Y be a continuous surjection of compact spaces X and Y satisfying the z.d.1.p. Then the following hold. (i) For any closed subset 2 of Y the induced restriction 6(19-‘(Z) : K’(Z) + 2 satisfies the z.d.1.p. . (ii) If c : X + 2 is an embedding of X into a compact spti:::e2 and 9: 2 + Y is a continuous extension of 8, i.e. $0 L = 8: then $ satisfies the z.d.1.p.

The following result on the notion of z.d.1.p. relies heavily on the fact that we are concerned with the lifting of maps defined on a zero-dimensional domain space. We show that the concept of z.d.1.p. is, by its nature, a local one.

o&ion 1.3. Let 8 : X + Y be a continuous surjection of compact spaces Xand Y. Suppose, we are given a family of closed subspaces ( Ya )acA of Ysuch that the union of 40 B. Hoffmann / An injective characterization of Peano spaces the interiors of the sets Ya(a E A) covers Y and a farnil) (Xo)ac~ of closed subspaces of X such that 8 maps Xa onto Y, for all a E A. Let 0,: X, + Y, (a E A) denote the restriction of 8 and assume &, satisfies the z. d.1.p. for all a E A. Then 8 satisfies the a.d.1.p.

Proof. Let T be a zero-dimensional compact space and 4 : T + Y a continuous map. Utilizing the z.d.1.p. of the maps & and the zero-dimensionality of the space T we shall construct a continuous lifting a: T + X for 4 by defining it locally. Let Ua denote the interior of Y, for all a E A. By a standard compactness argument there exists a finite subset F of A such that the union U {U, : a E F} coincides with Y. The continuity of 4 entails that the family (&-‘( &))dE~ is a finite, open covering of T. Since T is a zero-dimensional space there exists a finite refinement ( Tc)ecE of (&‘( &)) oeF consisting of pairwise disjoint, open (or, equivalently, closed) sets T, (e E E). Hence we can and do pick for each e E E an index a(e) E F such that Tc c 4-‘( U&cj) and consequently 4( T,) c U&) c Yacc). Restricting the map 4 to Te yields a continuous map &: T, + Yate) for all e E E. By hypothesis 8a(e) satisfies the z.d.1.p. and so we deduce the existence of a continuous lifting a,: Te + Xate) for 4e such that &(e) 0 0, = (bc (e E E). The pairwise disjointness of the clopen covering (Te)ecE o f T yields a continuous map o: T + X by setting crl Te:= we (e E E). Certainly, the identity 80 a =

We notice that in the hypotheses of the preceding proposition the covering condition cannot be relaxed to just requiring (Ya)(IE~ to cover Y. Consider the following example. Let X be the disjoint union Xi u X2 of two copies Xi, X2 of the one-point compactification of the integers. We take Y as the quotient space that we get by identifying the two points at infinity and 8: X + Y as the quotient map. Certainly, for i = 1,2 the restriction 8+ 8 ]Xi :Xi + 0(X,)=: Yi is the identity map and thus satisfies the z.d.1.p. Since, evidently, 8 is not a retraction and the image space Y of 8 is zero-dimensional we infer that 8 does not satisfy the z.d.1.p. In passing, we mention that Proposition 1.3 holds, if we substitute the term “satisfies the z.d.1.p.” by “allows an r.a.0.“; this is a consequence of Lemma 3.6 of Pelczyfiski’s monograph [8]; alternatively sl:e Lemma 1 of 123. The preceding proposition together with an observation made earlier yields that a continuous surjection of compact spices satisfies the z.d.1.p. if it is locally homeomorphic to a projection. Continuaus surjections of this type, often referred to as locally trivial fibrations, play an impor:*$antrole in the theory of algebraic topology. In this context, a well-known result of Gleason [3, Theorem 3.61 provides an interesting family of surjections satisfying the z.d.1.p. For instance, it implies that a quotient map of a compact modulo a closed Lie subgroup is a locally trivial fibration [3, Theorem 4.11 and hence satisfies the z.d.1.p. That the notion of z.d.1.p. is closed under the taking of continuous, well-ordered inverse limits is stated in the next proposition. Its straightforward proof, based on a B. Hoffmann / An injective characterization of Peano spaces 41 transfinite induction argument, immediately ff0110~s from a property of inverse limit spaces.

Proposition 1.4. Let (X0, p~rJp~B<7 be a well-ordered inverse system of compact spaces X, and continuous surjections pas :Xo +Xa ((r G p < T), indexed by the ordinals less than T and let X be the inverse lir& space lim(Xa, pno)uGpc8r. Suppose further that (i) for all limit ordinals y < T, the natural mapping from XY to @(X,, pcysJa

Proof. For convenience of notation we shall show that p1 satisfies the z.d.1.p. The proof for any other limit map pa (cy> 1) is pompletely analogous. So let T be a zero-dimensional compact space and 4: T -* XI a continuous map. By transfinite induction we shall construct a system of maps 0,: T +Xa for all Q!c r such that pago o8 = u, holds for all (YG p < r. Then by the property of the inverse limit space there exists a map a: T + X with pa 0 g = o-, for all cy< 7. We start the induction by putting cl := 4. Suppose we had defined m;aps a,: T + X, for all ac

We shall conclude this section with a result which arises out of an application of Propositions 1.3 and 1.4 and will be used as a crucial tool in the section to follow. A preparatory lemma is called for.

ILemma1.5. The unit interval I is the continuous image of the Cantor set D” by a map satisfying the z.d. 1.p.

Proof. We define a triadic tree of closed subintervals of I by the procedure of successive refinements. We start by choosing three subintervals X:“‘, X:“‘, X:“’ of 4 such that the length of X:” is ~2~’ and the union of the interiors of the sets X!? covers I. Repeating this process of refinement for each Xy’ instead of I yields closed subintervals Xy ‘) (j - 1,2,3) of Xy’ such that the length of Xki’ is ~2~~ and the union of the interiors (relative to X:i’) of Xg ‘) covers Xy’. Following along these lines a simple induction argument provides a family {,‘XF’: n 2 1, k ~{1,2,3}“} of closed subintcrvals of I such that for all ~rz3 1, k E {1,2, ,3}’ (a) for j = 1,2,3, Xlfr;i: is a closed subimerval of X’,“’ ; 42 B. Hoffmann / An injective characterization of Peano spaces

(b) the union of the interiors (relative to Xkk)) of XLk$‘:covers Xp’ ; (c) the length of the interval Xp’ is s2’“. For all integers II 2 0 let us define compact spaces Xn by taking X0 as the unit interval I and X, as the disjoint union of the intervals Xp’ Sor n a 1. The continuous surjection ~~,~+l of Xn+i onto Xn, assigning to each element of X n+l the same real number in Xn, satisfies the hypotheses of Proposition 1.3. For p m,n+lidentically takes the clopen subspace X’,“;i’ of Xn+l into a closed subspace of X, and by (b) and the definition of X,, the union of the interiors of the images pn,,,+l(X:2: ) = X’,“;‘:covers X,. Hence P,,,~+~satisfies the z.d.1.p. Putting pm”:=P~,~+~o l l + 0 ~~-1,~ for 0 s m G n < o yields a countable inverse system of compact spaces X, and continuous surjections pm”:X, +X,,, satisfying the assumptions of Proposition 1.4. Hence, if X denotes the inverse limit space &(X,, p,&mSn(W,we deduce that po :X 3 X0 = I satisfies the z.d.1.p.Employing (c) above, it is not difficult to see that any two distinct points of X can be separated by clopen sets. Hence X is a zero-dimensional, compact met&able space and therefore a retract of the Cantor set D”” [ l&8.3.4.]. Composing a retraction DU +.X with the map po: X * I yields the desired surjection D” + I satisfying the z.d.1.p D

Proposition 1.6. Let X be a compact space. Then there exists a zero-dimensional compact space T of the same weight as X and a continuous surjectiafn9: T +X satisfyingthe z.d.1.p.Furthermore, if Xis met&able, thenXis the continuousimage of D” by a map satisfyingthe z.d.1.p.

Prmf. If X is finite, everything is trivial. Therefore let us assume X to be of infinite weight 7. We consider X as a closed subspace of I*. By Lemma 1.5 th,ere exists a continuous surjection 4: D” + I satisfying the z.d.1.p. and so the canonical product map # := (&)cy47:(D”)’ -) I’, q&:= 4 for all Q!c T, satisfies the z.d.1.p. too. Now, statement @iJ of Lemma 1.2 implies the z.d.1.p. for the restriction 0:= # 1@-l(X) : e-‘(X) + X Obviously, T:= e-‘(X) is a zero-dimensional compact space such that the weight of X ano T coincides. Finally, if we are given a compact metrizable space X the inverse image space e-‘(X) is a zero-dimensional cornw\& metrizable space too and hence a retract of the Cantor set D” [lo, 8.3.4.). CorNpositionof a retraction DW-+ e’*(X) with the restriction 8: e-‘(X) + X provides the desired surjection D”’ + X siatisfying the z.d.1.p. a

A technique somewhat similar to the one exhibited in the proof of Proposition 1.6 was employed by S.Z. Ditor in [2] t3rprove his Theorem 1. However, it is not hard to show that Ditor’s theorem is actually equivalent to Milutin’s lemma, stating that the unit interval %is the continuous image of the Cantor set D” by a map admitting an r.a.0. In fact, it follows from the results of [SJ, quoted earlier, that Lemma 1.5 and B. Hoffmann / An injective characterization of Peano spaces 43

Milutin’s lemma are equivalent too. But in contrast to all known proofs of Milutin’s lemma our proof of Lemma 1.5 is of a purely topological nature and comparatively brief. So we gave it. Proposition 1.6 can be considered as an extension of Milutin’s lemma in topological terms, not involving the algebraic notion of r.a.o.

2, AE(0,m) and Peano spaces

Having developed all the necessary tools in the preceding section we shall now be able to furnish our main results with fairly short proofs. Before restricting our attention to met&able spaces we utilize Proposition 1.6 to give a surjective characterization of the injectively defined class of spaces of type AE(0, a).

Theorem 2.1. For a compact space X the following are equitralent. (1) X is an AE(0, a~). (2) X is the continuous image of some Tychonoff cube I’ by a map satisfying the 2.d. 1.p.

Proof. If X i_;a finite space, both statements (1) and (2) imply that X consists of a single point. %o we may assume X to be of infinite weight 7. To prove tltat (1) entails (2) we employ Proposition 1.6 and infer the existence of a zero-dimensional compact space T of weight 7 and a continuous surjection q5

4

rS commutes. Thus 80 # is a continuous extensioen of 4. 0

Rephrasing the result of Theorem 2.1 yields the following characterization. The class of spaces of type AE(0,00) is the smallest class of compact spaces that contaiins 44 ~B, Hoffmann/ An injective characterizationof Peano spaces the unit interval I and is closed under the operations of taking arbitrary products and continuous images by maps satisfying the z.d.1.p. In passing, we remark that the statement of Theorem 2.1 completely corresponds to the surjective characterization of Dugundji spaces we established in [S]. There an AE(0,0) or, equivalently, by Haydon’s result [4], a Dugundji space is shown to be the continuous image of a Cantor set D’ by a map satisfying the z.d.1.p. Concentrating on compact met&able spaces the results of Section 1 togethlerwith the preceding theorem yield the following equivalences.

Theorem 2.2. For a compact space X the following are equivalent. (1) X is a met&able AE(0,00). (2) X is a met&able AE(0, 1). (3) X is a continuous image of the unit interval. . (4) X is a Peano space. (5) X is a continuous image of the unit interval by a map satisfying the zd.1.p. (6) X is a continuous image of the unit interval by a map allowing an r.a.0.

Proof. Certainly, without restriction to metrizability (1) implies (2). That (3) follows from (2) is an easy consequence of the well-known fact that a compact met&able space is the image of Do by a continuous map; then by hypothesis this surjection allows a continuous extension to the one-dimensional space 1. The equivalence of the statements (3) and (4) is the well-known Hahn-Mazurk- iewicz theorem [13, Theorem 31.51. To prove that (4) implies (5) we recall the standard method that is employed to deduce (3) from (4) (see e.g. [13, Proof of Theorem 3 1.51).One takes any continuous surjection D” + X and utilizes the connectivity and local connectivity of the metriz- able space X to continuously extend this surjection to a map I + X. Now, instead of starting with an arbitrary map of Dw onto X we may choose a continuous surjection D” 3 X satisfying he z.d.1.p. by Proposition 1.6. As just explained the assumption (4) guarantees the existence of a continuous ension I +X lvhich then satisfies the z.d.1.p. too. This follows from statement (Fi) Lemma 1.2. The equivalence of the statements (5) and 46) is a consequence of a result of [S], quoted in the preceding section. That (6) or, equiv&ntly, (5) implies (1; we deduce from Theorem 2.1. cl

In Example 2 of his monograph [S, p. 651 Pelczyfiski showed that the map of the unit interval I onto the one-dimensional unit circle S”, taking an element t of I into e 2”it,does not allow an r.a.o. If S” denotes the n-dimensional unit sphere, then, for all n 2 1, Theorem 2.2 implies the existence {ofa continuous surjection of I onto S” allowing an r.a.o. For S” is known to be a metrizable AE(n, n) [7, Theorem 8-191and hence, in particular, an AE(0,1). 61. Hoffmann / An injective characterizata’onof Peano spaces 45

Bearing in mind that a Peano space is a complete metric space which is locally arcwise connected and arcwise connected (see [13, Theorems 31.2 and 31.41) and hence, in particular, a space of type LC” and Co simultaneously, an application of an extension theorem of E. Michael [6, Theorem 1. l] together with the preceding theorem yields the following characterization. Although we believe that this result is well-known to the specialist we choose to include it in this paper.

Ptopos&n 2.3. A compact met&able space is a Peano space if and only if it is an AE(l, 1).

Note that the one-dimensional unit circle S’ is an AE( 1,l) but certainly not an AE(l, 2), since it is not an retract of the two-dimensional unit square 12. Therefore Proposition 2.3 does not allow an improvement. Finally, we mention that/in general, a locally connected, connected AE(0,0) of weight o1 is not an AE(0, a~). For a compact space X we denote by exp(X) the space of non-empty, closed subsets of X, endowed with the finite topology. It follows from Corollary 4 and Theorem 5 of [9] that the space exp(l”l) is a locally connected, connected Dugundji space and not the continuous image of a Tychonoff cube. Hence by Haydon’s characterization of Dugundji spaces as AE(0,0) and Theorem 2.1, the space exp(loh*) is a locally connected, connected AE(0,0), but not an AE(0, a~). In this context, it seems interesting to recall Theorem 1 of [l]. A compact space X is a non-degenerate Peano space if and only if exp(X) is homeomorphic to IO.

Acknowledgement

I should like to thank my supervisor Dr. R.G. Haydon for introducing me to the notion of z.d.1.p. and for bringing Michael’s ipaper [6] to my attention. Furthermore, I am grateful to the referee for asking for the characterization of Peano spaces given in Proposition 2.3.

References

113 D-W. Curtis and R.M. Schori, 2x and C(X) are homeomorphic to the Hilbert cube, Bull. Amer. Math. Sot. 80 (1974) 927-931. 12) S.Z. Ditor,, On a lemma of Milutin concerning averaging operators in spaces, Trans. Amer. Math. Sot. 149 (1970) 443-452. [3] A.M. Glea:i;on,Spaces with a compact Lie group of transformation, Proc. Amer. Math. Sot. l(1950) 3543 I [41 R. Haydon, On a problem of Peiczyiiski: Milutiarspaces, Dugundji spaces and AE(O-dim), Studia Math. 52 (1974) 23-31. [5] B. Hoffmann, A surjective characterization of Dugundji spaces, Proc. Amer. Math. Sot. 76 (1979) 151-156. [63 E. Michael, Continuous selections II, Ann. of M;ath.64 (1956) 562-580. [7] .K. Nagami, Dimension theory (Academic Press, New York and London, 1970). 46 B. Ho&wan / Arr injecfiue characterization of Peano spaces

[8] A. Pelezyriski, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Diss. Math. 58, Warszawa, 1968. [9] L.B. Sapiro, On spaces of closed subr;ets of bicompacts, Soviet Math. Dokl. 17 (1976) 1567-1571. [ 10] 2. Semadeni, Banach spaces of continuous functions (Polish Scientific Publishers, Warszawa, 1971). [l l] E. D. Tymchatym, The Hahn-Mazurkiewicz theorem for finitely Suslinian continua, Gen. Topology Appl. 7 (1977) 123-128. fl2] L.E. Ward, Jr., The Hahn-Mazurkiewicz theorem for rim-finite continua, Gen. Topology J?.ppl.6 (1968) 183-190. [13] S. WiiEard,Topology (Addison-Wesley, Reading, 1971).